Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist.[3][4] Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.

An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique.[1]

The term arborescence comes from French.[5] Some authors object to it on grounds that it is cumbersome to spell.[6] There is a large number of synonyms for arborescence in graph theory, including directed rooted tree[2][6]out-arborescence,[7]out-tree,[8] and even branching being used to denote the same concept.[8]Rooted tree itself has been defined by some authors as a directed graph.[9][10][11]

Furthermore, some authors define an arborescence to be a spanning directed tree of a given digraph.[11][12] The same can be said about some its synonyms, especially branching.[12] Other authors use branching to denote a forest of arborescences, with the latter notion defined in broader sense given at beginning of this article,[13][14] but a variation with both notions of the spanning flavor is also encountered.[11]

It's also possible to define a useful notion by reversing all the arcs of an arborescence, i.e. making them all point to the root rather than away from it. Such digraphs are also designated by a variety of terms such as in-tree[15] or anti-arborescence[16] etc. W. T. Tutte distinguishes between the two cases by using the phrases arborescence diverging from [some root] and arborescence converging to [some root].[17]

The number of rooted trees (or arborescences) with n nodes is given by the sequence: